OptBand: optimization-based confidence bands for functions to characterize time-to-event distributions

Classical simultaneous confidence bands for survival functions (i.e., Hall-Wellner, equal precision, and empirical likelihood bands) are derived from transformations of the asymptotic Brownian nature of the Nelson-Aalen or Kaplan-Meier estimators. Due to the properties of Brownian motion, a theoretical derivation of the highest confidence density region cannot be obtained in closed form. Instead, we provide confidence bands derived from a related optimization problem with local time processes. These bands can be applied to the one-sample problem regarding both cumulative hazard and survival functions. In addition, we present a solution to the two-sample problem for testing differences in cumulative hazard functions. The finite sample performance of the proposed method is assessed by Monte Carlo simulation studies. The proposed bands are applied to clinical trial data to assess survival times for primary biliary cirrhosis patients treated with D-penicillamine.

An empirical likelihood method for semiparametric linear regression with right censored data

This paper develops a new empirical likelihood method for semiparametric linear regression with a completely unknown error distribution and right censored survival data. The method is based on the Buckley-James (1979) estimating equation. It inherits some appealing properties of the complete data empirical likelihood method. For example, it does not require variance estimation which is problematic for the Buckley-James estimator. We also extend our method to incorporate auxiliary information. We compare our method with the synthetic data empirical likelihood of Li and Wang (2003) using simulations. We also illustrate our method using Stanford heart transplantation data.

Visualizing Multiple Quantile Plots

Multiple-quantile plots provide a powerful graphical method for comparing the distributions of two or more populations. This article develops a method of visualizing triple-quantile plots and their associated confidence tubes, thus extending the notion of a quantile-quantile (QQ) plot to three dimensions. More specifically, we consider three independent one-dimensional random samples with corresponding quantile functions Q₁, Q₂, and Q₃. The triple-quantile (QQQ) plot is then defined as the three-dimensional curve Q(p) = (Q₁(p), Q₂(p), Q₃(p)), where 0 < p < 1. The empirical likelihood method is used to derive simultaneous distribution-free confidence tubes for Q. We apply our method to an economic case study of strike durations and to an epidemiological study involving the comparison of cholesterol levels among three populations. These data as well as the Mathematica code for computation of the tubes are available in the online supplementary materials.

Confidence intervals for survival quantiles in the Cox regression model

Median survival times and their associated confidence intervals are often used to summarize the survival outcome of a group of patients in clinical trials with failure-time endpoints. Although there is an extensive literature on this topic for the case in which the patients come from a homogeneous population, few papers have dealt with the case in which covariates are present as in the proportional hazards model. In this paper we propose a new approach to this problem and demonstrate its advantages over existing methods, not only for the proportional hazards model but also for the widely studied cases where covariates are absent and where there is no censoring. As an illustration, we apply it to the Stanford Heart Transplant data. Asymptotic theory and simulation studies show that the proposed method indeed yields confidence intervals and bands with accurate coverage errors.